Manipulating sets of hierarchical data

ABSTRACT

Embodiments of methods, apparatuses, devices and/or systems for manipulating hierarchical sets of data are disclosed.

BACKGROUND

This disclosure is related to hierarchical data arrangements and, moreparticularly, to manipulating such data arrangements.

In a variety of fields, data or a set of data, may be represented in ahierarchical fashion. This form of representation may, for example,convey information, such as particular relationships between particularpieces of data and the like. However, manipulating such datarepresentations is not straight-forward, particularly where the data isarranged in a complex hierarchy. Without loss of generality, one examplemay include a relational database. Techniques for performing operationson such a database, for example, are computationally complex orotherwise cumbersome. A continuing need, therefore, exists foradditional techniques for manipulating data hierarchies.

BRIEF DESCRIPTION OF THE DRAWINGS

Subject matter is particularly pointed out and distinctly claimed in theconcluding portion of the specification. The claimed subject matter,however, both as to organization and method of operation, together withobjects, features, and advantages thereof, may best be understood byreference of the following detailed description when read with theaccompanying drawings in which:

FIG. 1 is a schematic diagram of one embodiment of a tree;

FIG. 2 is a schematic diagram illustrating one embodiment of a binaryedge labeled tree;

FIG. 3 is a schematic diagram illustrating another embodiment of abinary edge labeled tree;

FIG. 4 is a schematic diagram illustrating an embodiment of a binaryedge labeled string;

FIG. 5 is a table illustrating representations of binary strings andbinary numerals that may be associated in one particular embodiment;

FIG. 6 is a table illustrating representations of binary strings thatmay be associated in one particular embodiment;

FIG. 7 is another table illustrating representations binary strings thatmay be associated in one particular embodiment;

FIGS. 8 a, 8 b, and 8 c are a table illustrating a particular embodimentof an association between binary stings and binary edge labeled trees;

FIG. 9 is a table illustrating a portion of the Kleene enumeration ofnon-composite numerals;

FIG. 10 is a table illustrating one embodiment of a linear notation forrepresenting a graphical depiction of a hierarchical set of data;

FIG. 11 is a schematic diagram of an embodiment of a node labeled tree;

FIGS. 12-17 illustrate application of various operations to convert theembodiment of FIG. 11 from one type of tree to another type of tree;

FIG. 18 is a schematic diagram illustrating another embodiment of a nodelabeled tree;

FIGS. 19-22 illustrate application of various operations to convert theembodiment of FIG. 18 from one type of tree to another type of tree; and

FIG. 23 is a schematic diagram illustrating one embodiment of atechnique for combining binary strings.

DETAILED DESCRIPTION

In the following detailed description, numerous specific details are setforth to provide a thorough understanding of the claimed subject matter.However, it will be understood by those skilled in the art that theclaimed subject matter may be practiced without these specific details.In other instances, well-known methods, procedures, components and/orcircuits have not been described in detail so as not to obscure theclaimed subject matter.

Some portions of the detailed description which follow are presented interms of algorithms and/or symbolic representations of operations ondata bits or binary digital signals stored within a computing systemmemory, such as a computer memory. These algorithmic descriptions and/orrepresentations are the techniques used by those of ordinary skill inthe data processing arts to convey the substance of their work to othersskilled in the art. An algorithm is here, and generally, considered tobe a self-consistent sequence of operations and/or similar processingleading to a desired result. The operations and/or processing involvephysical manipulations of physical quantities. Typically, although notnecessarily, these quantities may take the form of electrical and/ormagnetic signals capable of being stored, transferred, combined,compared and/or otherwise manipulated. It has proven convenient attimes, principally for reasons of common usage, to refer to thesesignals as bits, data, values, elements, symbols, characters, terms,numbers, numerals and/or the like. It should be understood, however,that all of these and similar terms are to be associated with theappropriate physical quantities and are merely convenient labels. Unlessspecifically stated otherwise, as apparent from the followingdiscussion, it is appreciated that throughout this specificationdiscussions utilizing terms such as “processing”, “computing”,“calculating”, “determining” and/or the like refer to the actions and/orprocesses of a computing platform, such as a computer or a similarelectronic computing device, that manipulates and/or transforms datarepresented as physical electronic and/or magnetic quantities within thecomputing platform's memories, registers, and/or other informationstorage, transmission, and/or display devices.

In a variety of fields, data or sets of data may be represented in ahierarchical fashion. This form of representation may, for example,convey information, such as particular relationships between particularpieces of data and the like. However, manipulating such datarepresentations is not straight forward, particularly where the data isarranged in a complex hierarchy. Without loss of generality, one examplemay include a relational data base. Techniques for performing operationson such a data base for example, may be computationally complex orotherwise cumbersome. A continuing need, therefore, exists foradditional techniques for manipulating data hierarchies.

As previously discussed, in a variety of fields, it is convenient ordesirable to represent data, a set of data and/or other information in ahierarchical fashion. In this context, such a hierarchy of data shall bereferred to as a “tree.” In a particular embodiment, a tree may comprisea finite, rooted, connected, unordered, acyclic graph. This isillustrated here, for example, in FIG. 1 by embodiment 100. Asillustrated, the root of this particular embodiment encompasses node105. In addition to 105, there are eight other nodes designated 110 to140, respectively. Likewise, the nodes are connected by branchesreferred to, in this context, as edges. Thus, the nodes of this tree areconnected by eight edges. This embodiment, therefore, illustrates afinite tree that is rooted by node 105. Furthermore, the nodes areconnected, meaning, in this context, that a path exists between any twonodes of the tree. The tree is likewise acyclic, meaning here, that nopath in the tree forms a complete loop. Here, unordered refers to thenotion that there is no implied ordering or precedence among nodesattached to a common node, despite the appearance of ordering in agraphical illustration.

As previously suggested, in a variety of contexts, it may be convenientand/or desirable to represent a hierarchy of data and/or otherinformation using a structure, such as the embodiment illustrated inFIG. 1. One particular embodiment, without loss of generality, of a treemay include edges that are labeled with data and/or other values.Likewise, in one particular embodiment, such data or values may belimited to binary data, that is, in this example, either a binary one ora binary zero. Here, such an embodiment may be referred to as a binaryedge labeled tree (BELT), as shall be discussed in more detailhereinafter.

One example of a BELT is illustrated by embodiment 200 of FIG. 2. Thus,as illustrated, the edges of the BELT shown in FIG. 2 are labeled witheither a binary zero or binary one. FIG. 3 illustrates anotherembodiment 300 of a different binary edge labeled tree. It is noted thatthis tree is similar or isomorphic to the embodiment of FIG. 2.

A subset of BELTs may be referred to in this context as binary edgelabeled strings (BELSs). One embodiment, 400, is illustrated in FIG. 4.Thus, as illustrated by embodiment 400, this particular binary edgelabeled string comprises four nodes and three edges, where the edges arelabeled, respectively, binary zero, binary one and binary zero. Thus, abinary edge labeled string comprises a binary edge labeled tree in whicheach node has no more than two edges. To reiterate, in this context, astring comprises a binary edge labeled string and a tree comprises abinary edge labeled tree if each edge of the string or tree respectivelystores a single bit. Likewise, in this context, two nodes are employedto support an edge holding a single piece of binary data. At this point,it is worth noting that strings and trees having nodes and edges, suchas previously described, may be represented in a computing platform orsimilar computing device through a data structure or a similar mechanismintended to capture the hierarchical relationship of the data. It isintended that all such embodiments are included within the scope of theclaimed subject matter.

As may be apparent by a comparison of FIG. 4 with, for example, FIG. 2or FIG. 3, typically a binary edge labeled tree has the ability to bericher and convey more data and/or more information than a binary edgelabeled string, although, of course, depending on the particular treeand the particular string, there may be contrary examples, such as wherethe string is particularly large and the tree is particular small. Theaspect of BELTs to be richer in information may be one potentialmotivation to employ BELTs over BELSs, for example.

Despite the prior observation, as shall be described in more detailhereinafter, an association may be made between any particular binaryedge labeled string and a binary edge labeled tree or vice-versa, thatis, between any particular binary edge labeled tree and a binary edgelabeled string. In particular, as shall be explained in more detailhereinafter, an association may be constructed between binary edgelabeled trees and binary edge labeled strings by enumerating in aconsecutive order binary edge labeled strings and binary edge labeledtrees, respectively, and associating the respectively enumerated stringsand trees with natural numerals. Of course, as shall become more clearhereinafter, many embodiments of associations between trees and stringsor between trees and natural numerals are possible. It is intended thatthe claimed subject matter include such embodiments.

In this context, it may be useful to draw a distinction between binarynumerals or binary data values and binary strings. FIG. 5, for example,illustrates a binary string and its associated binary numeral or datavalue. All binary numerals or binary data values may be represented witha binary string, but not vice-versa. One reason for this is that allbinary data values or numerals, except the binary data value zero, beginwith a binary one, but not so for binary strings. Likewise, there aremany different ways to represent binary strings, and the claimed subjectmatter is not limited to any particular technique to make such arepresentation. For example, FIG. 6 illustrates two techniques orapproaches to viewing essentially or substantially the same binarystring.

FIG. 6, therefore, has three columns. Column two contains theconventional or standard method to represent a binary string, such asmay be employed in the field of computer science, for example. Along thesame row, column three contains a graphical representation ofsubstantially the same binary string using a binary edge labeled string,such as previously described. Likewise, column one, along the same row,contains a natural numeral that corresponds to the particular binarystring of the particular row.

However, this particular order of binary strings is not the typicalorder that is well-known and employed, for example, in technical fields,such as computer science and/or electrical engineering. Typically, forbinary numerals, for example, to form the next consecutive binarynumeral after one, a binary digit or bit is added to the right. Forexample, in an ordered sense, the binary numeral following binarynumeral one, (1)₂, is the binary numeral one zero, (10)₂.

In contrast, for the table illustrated in FIG. 6, a binary digit or bitis added to the left; however, a similar convention of adding a binaryzero before a binary one is employed. Thus, as one example, the binarystring after the binary string, one <1>, is the binary string, zero zero<0,0>. Furthermore, this particular ordering or enumeration of binarystrings begins with the string with no nodes, which, for this particularembodiment, is associated with the natural numeral zero and has asymbolic representation here that comprises a lambda (λ). In thiscontext, this may be referred to as the empty binary string or zero nodebinary edge labeled string. Next in the enumeration is the binary stringthat comprises a single node and, therefore, holds no data. In thisparticular embodiment, this is associated with the natural number one,and is depicted by a graphical representation that comprises a singlenode. This is referred to here as the one node binary string. For higherpositive natural numerals, as previously suggested, to represent asingle bit for a binary edge labeled string, two nodes are employed.

One technique for converting from a particular binary string, such as incolumn two, for example, to the natural numerals in column one, includesinserting a binary numeral one in front of the binary string, and thenconverting the binary numeral one plus the binary digits of the binarystring to a binary numeral. The natural numeral, corresponding to thatbinary numeral by converting from base two to base ten, provides thedesired result. Thus, one embodiment of a method of enumerating a set ofstrings, in this case binary strings, includes positioning, at alocation k, where k represents a positive natural numeral, a binarystring such that the string comprises the binary numeral correspondingto k with the left most binary digit omitted. Again, as previouslysuggested and illustrated in FIG. 6, by column three, binary strings maybe represented by binary edge labeled strings or BELS, although theclaimed subject matter is not limited in scope in this respect.

As alluded to previously, there are many ways to represent binarystrings and it is intended to include all such representations withinthe scope of the claimed subject matter. As simply one example, FIG. 7illustrates yet another view of a binary string. Column two of FIG. 7provides particular examples of binary node labeled strings, rather thanbinary edge labeled strings. Thus, as will be discussed in more detailhereinafter, binary edge labeled strings and binary node labeled stringsmay comprise nearly interchangeable representations of a binary string,although, again, other representations are also included within thescope of the claimed subject matter.

In addition to enumerating binary strings, which may be accomplished byadding a bit to the left, as previously described, in one embodiment,alternatively, if a set of specific binary strings are provided, thosestrings may be ordered. An embodiment of a method of ordering, such asfor the previously described embodiment, includes proceeding longerstrings with shorter strings. However, for ordering strings of the samelength, such strings may be ordered by converting to their associatedvalues, as previously described, by adding binary numeral one to theleft of the string, and placing the associated values in conventionalascending numerical order.

Just as binary strings may be ordered and/or enumerated, likewise binaryedge labeled trees may also be enumerated and/or ordered. This isillustrated, for example, in FIG. 8. In this particular table, two viewsor representations of binary strings are associated with a particularrepresentation of binary edge labeled trees. It is noted that thisparticular figure also includes the associated natural numerals. Theassociation of such numerals for this particular embodiment should beclear based at least in part on the prior description.

Thus, for this particular embodiment, although the claimed subjectmatter is not limited in scope in this respect, a method of enumeratinga set of trees begins with enumeration of an empty binary edge labeledtree and a one node binary edge labeled tree, similar to the emptybinary string and one node binary string, previously described. Thus, asfor binary strings, here, the empty tree is associated with the naturalnumeral zero and has a symbolic representation as illustrated in FIG. 8a (circle). Likewise, the one node tree, which holds no data, isassociated with the natural numeral one and has a graphicalrepresentation of a single node. For higher positive natural numbers,however, this embodiment of a method of enumerating a set of treescomprises positioning a tree at location k, k being a positive naturalnumeral greater than three, where k is the product of u and v, u and vcomprising positive natural numerals greater than one, such that thetree is formed by a union of the trees at positions u and v. Likewise,for those locations that are not a product of other natural positivenumerals greater than one, that is, for locations that comprisenon-composite numerals, denoted here by j, for example, j being apositive natural number greater than one, a tree is positioned atlocation j such that the tree is formed by finding the first tree in theprior enumeration such that the binary edge labeled tree obtainable fromthis first tree by attaching a node to the particular tree as a new rootnode and labeling the edge between the new root node and the prior rootnode with a binary “0” label is not in the enumeration at some positionlower than j; however, if the binary edge labeled tree obtainable fromthat first tree, as just described, is present in the enumeration with abinary “0” label for the new edge, but not with a binary “1” label, thenthe tree at position j is that tree with a binary “1” label for the newedge. This may be illustrated, for example in FIG. 8, as described inmore detail below.

As illustrated, for this particular embodiment, and as previouslydescribed, the empty tree has zero nodes and is associated with thenatural numeral zero. Likewise, the one node tree root comprises asingle node and is associated with the natural numeral one. Thus, toobtain the tree at position two, a root node is attached and connectedto the prior root node by an edge. Likewise, here, by convention, theedge is labeled with a binary zero. If, however, the tree formed by theimmediately proceeding approach were present in the prior enumeration oftrees, then a similar process embodiment is followed, but, instead, thenew edge is labeled with a binary one rather than a binary zero. Thus,for example, in order to obtain the binary edge labeled tree forposition three, a new root node is connected to the root node by an edgeand that edge is labeled with a binary one.

Continuing with this example, to obtain the binary edge labeled tree forposition four, observe that numeral four is the product of numeral twotimes numeral two. Thus, a union is formed at the root of two trees,where, here, each of those trees is associated with the positive naturalnumeral two. Likewise, to obtain the binary edge labeled tree forposition five, begin with the binary edge labeled tree for position twoand follow the previously articulated approach of adding a root and anedge and labeling it with a binary zero.

In this context, adding a root node and an edge and labeling it binaryzero is referred to as a “zero-push” operation and adding a root nodeand an edge and labeling it binary one is referred to as a “one-push”operation. Based at least in part on the prior description, for thisparticular embodiment, it may now be demonstrated that if k is anypositive natural numeral and a tree x is positioned at location k, thena non-composite numeral is associated with the zero-push of that treeand a non-composite numeral is associated with the one-push for thattree. Furthermore, the non-composite index of the zero-push of the treecomprises 2k−1, whereas the non-composite index of the one-push of thetree comprises 2k, where the index corresponds to the argument of thewell-known Kleene enumeration on positive natural numerals ofnon-composite numerals, as illustrated, for example, in part in FIG. 9.Thus, referring again to FIG. 8, the one-push of the root tree is thetree at position three. This follows from FIG. 9 since P(2*1)=P(2)=3.Likewise, the tree at position five is the zero-push of the tree atposition 2. Again, this follows from FIG. 9 since P(2*2−1)=P(3)=5.

In this context, the approach just described may be referred to asvectorizing non-composite numerals. In the embodiment just described,this was accomplished in pairs, although, of course, the claimed subjectmatter is not limited in scope in this respect. This may be accomplishedin any number of numeral combinations, such as triplets, quadruplets,etc. Thus, using a quadruplet example, it is possible to construct treessuch that if k is any positive natural numeral and a tree x ispositioned at location k, then a non-composite numeral is associatedwith the zero-push of that tree, a non-composite numeral is associatedwith the one-push for that tree, a non-composite numeral is associatedwith the two-push for that tree, and a non-composite number isassociated with the three-push for that tree. Furthermore, the index ofthe non-composite numeral is such that for a zero-push of the tree, theindex comprises (4k−3), for a one-push of a tree, the index comprises(4k−2), for a two-push of a tree, the index comprises (4k−1), and for athree-push of a tree the index comprise (4k), where the indexcorresponds to the Kleene enumeration of non-composite numerals,P(index), such as provided in FIG. 9.

In the previously described enumeration of binary edged labeled trees, amechanism may be employed to reduce or convert complex manipulations ofhierarchical data to multiplication of natural numerals. For example, ifit is desired to combine, or merge at their roots, two trees ofhierarchical data, a complex task both computationally and graphically,instead, for this particular embodiment, the two trees may be convertedto numerical data by using the previously described associationembodiment between binary edge labeled trees and natural numerals. Theresulting numerical data from the prior conversion may then bemultiplied, and the resulting product may then be converted to a binaryedge labeled tree by using a table look up of the previously describedassociation embodiment. It is noted that a subtle distinction may bemade between an enumeration embodiment and an association embodiment.Enumeration may comprise listing, in this example, a particular orderedembodiment of BELTs, whereas an association provides a relationshipbetween, in this example, a particular ordered embodiment of BELTs andnatural numerals. It is, of course, appreciated that many differentenumeration and association embodiments may be employed to execute theoperations discussed above and hereinafter, and the claimed subjectmatter is intended to cover all such enumeration and associationembodiments.

Alternatively, a similar approach may be employed to combine two treesusing binary strings, rather than numerical data. Thus, using the priorembodiment previously discussed in which particular binary strings andBELTs are associated, the BELTs may be converted to binary strings. Thebinary strings may be combined, and the resulting combination of stringsmay then be converted to a tree using the association embodiment, asdescribed, for example, above. It is noted that in this particularcontext, combining binary strings refers to an operation as illustratedin FIG. 26. The binary strings are converted to binary numerals, such aspreviously described. The converted binary numerals may be multiplied,and the resulting binary numeral product may then be converted to abinary string, again, as previously described.

Likewise, a process embodiment that is a reversal to the previouslydescribed embodiments may also be employed. Thus, complex hierarchies ofdata may be split or divided, when this is desired. For example, abinary edge labeled tree to be divided may be converted to a piece ofnumerical data, such as by using the previously described associationembodiment. This data may then be factored into two pieces of numericaldata whose product produces the previously mentioned piece of numericaldata. These two pieces of numerical data may then be converted to trees,again, by using the prior association embodiment, for example.

A similar approach may be employed using binary strings, analogous tothe approach described above using binary strings to combine trees.Thus, a tree to be divided may be converted to a binary string using,for example, the previous association embodiment. This string may besplit into two separate binary strings and these two separate binarystrings may then be converted to two binary edge labeled trees, usingthe association embodiment previously discussed. Again, to do so, thebinary string may be converted to a binary numeral. The binary numeralmay then be factored into two binary numerals, and these two binarynumerals may be converted to binary strings.

Another form of manipulating hierarchical sets of data may involveordering or hashing. This may be desirable for any one of a number ofdifferent operations to be performed on the sets of data. One approachis similar to the previously described embodiment. For example, it maybe desired to order a given set of trees. Doing so may involveconverting the trees to numerical data, as previously described, usingan association embodiment. The numerical data may then be ordered andthe numerical data may then be converted back to binary edge labeledtrees using the previously described association embodiment, or analternate association embodiment, for example.

It is noted that there may be any one of a number of different ways ofconverting from numerals or numerical data values to a binary edgelabeled tree or from a binary string to a binary edge labeled tree, andvice-versa. Nonetheless, a convenient method for doing so with thisparticular embodiment includes storing a table providing an associationembodiment between natural numerals, binary strings and binary edgelabeled trees, such as the embodiment previously described. Thus, onceit is desired to convert from one to the other, such as from a binarystring to a BELT, from a natural number to a BELT, or vice-versa, forexample, a table look up operation may be performed using theassociation embodiment.

Techniques for performing table look ups are well-known andwell-understood. Thus, this will not be discussed in detail here.However, it shall be appreciated that any and all of the previouslydescribed and/or later described processing, operations, conversions,transformations, manipulations, etc. of strings, trees, numerals, data,etc. may be performed on one or more computing platforms or similarcomputing devices, such as those that may include a memory to store atable as just described, although, the claimed subject matter is notnecessarily limited in scope to this particular approach. Thus, forexample, a hierarchy of data may be formed by combining two or morehierarchies of data, such as by applying a previously describedembodiment. Likewise, multiple hierarchies of data may be formed bysplitting or dividing a particular hierarchy of data, again, such as byapplying a previously described embodiment. Likewise, additionaloperations and/or manipulations of data hierarchies may be performed,such as ordering hierarchies of data and more. It is intended that theclaimed subject matter cover such embodiments.

Much of the prior discussion was provided in the context of binary edgelabeled trees. Nonetheless, as alluded to previously, binary edgelabeled trees and binary node labeled trees may be employed nearlyinterchangeably to represent substantially the same hierarchy of data.In particular, a binary node labeled tree may be associated with abinary edge labeled tree where the nodes of the binary node labeled treetake the same values as the edges of the binary edge labeled tree,except that the root node of the binary node labeled tree may comprise anode having a zero value or a null value. This is illustrated, forexample, in FIG. 7. Thus, rather than employing binary edge labeledtrees, the previously described embodiments may alternatively beperformed using binary node labeled trees. As one example embodiment,operations and/or manipulations may be employed using binary edgelabeled trees and then the resulting binary edge labeled tree may beconverted to a binary node labeled tree. However, in another embodiment,operations and/or manipulations may be performed directly using binarynode labeled trees where a different association embodiment, that is, inthis example, one that employs binary node labeled trees, is employed.

In accordance with the claimed subject matter, therefore, any tree orany string, regardless of whether it is binary edge labeled, binary nodelabeled, non-binary, a feature tree, or otherwise, may be manipulatedand/or operated upon in a manner similar to the approach of thepreviously described embodiments. Typically, different associationembodiments shall be employed, depending at least in part, for example,upon the particular type of tree and/or string. For example, and asshall be described in more detail below in connection with FIG. 11, anode labeled tree in which the nodes are labeled with natural numeralsor data values may be converted to a binary edge labeled tree.Furthermore, this may be accomplished with approximately the same amountof storage. For example, for this particular embodiment, this mayinvolve substantially the same amount of node and/or edge data labelvalues.

As previously noted, the claimed subject matter is not limited in scopeto this particular example, however, as illustrated in more detailhereinafter, the tree illustrated in FIG. 11 is converted to a binaryedge labeled tree through a sequence of processing depicted here asgraph operations, although such a conversion may alternatively beimplemented by operations implemented otherwise, one such example beinga computing platform, for example. Alternatively, it may be desirable,depending upon the particular embodiment, to convert trees to, forexample binary node labeled trees. Likewise, other embodiments in whichtrees of one form are converted to trees of another form are alsoincluded within the scope of the claimed subject. However, for thisparticular embodiment, it will be assumed that the association betweentrees and strings, such as previously described, is depicted orenumerated in terms of binary edge labeled trees, as previouslyillustrated, for example. Thus, in this example, a particular tree,embodiment 1100, is illustrated in FIG. 11, comprises a node labeledtree rather than an edge labeled tree.

Referring now to FIG. 11, node values are deleted for those nodesstoring the value zero. This is illustrated, for example, in FIG. 12.Node values are also deleted for the nodes storing the value one. Inthis case, a new single unlabeled node is attached to those nodes thathad stored the value one, and the edge between the prior node and thenew node is labeled with a zero value. This is illustrated, for example,in FIG. 13. For nodes with a value k that is a power of 2, and largerthan one, the node value is deleted and (log₂(k)+1) new nodes areattached, labeling the edge between the new nodes and the old nodes witha zero value. This, for example, is illustrated in FIG. 14.

The remaining node values comprise non-powers of two that are 3 orlarger. These node values are factored into one or more non-compositenumerals. For the resulting non-composite numerals, the non-compositenumeral is replaced with the tag value of the index, for example, i, forthe non-composite. For this particular embodiment, the term, tag valueof index, i, for example, refers to a binary edge labeled tree in thepreviously discussed embodiment of a string-tree association thatcorresponds to a binary string associated with the binary numeral for i.The new edges of the tree are labeled with the binary value zero. Thisis illustrated, for example, in FIG. 15. The remaining edges of the treeare labeled with a binary value of one. This is illustrated in FIG. 16.

In another embodiment, however, a particular tree may include null typesor, more particularly, some node values denoted by the empty set. Thisis illustrated, for example, by the tree in FIG. 17, although, ofcourse, this is simply one example. An advantage of employing null typesincludes the ability to address a broader array of hierarchical datasets. For example, without loss of generality and not intending to limitthe scope of the claimed subject matter in any way, a null type permitsrepresenting in a relational database, as one example, situations wherea particular attribute does not exist. As may be appreciated, this isdifferent from a situation, for example, where a particular attributemay take on a numeral value of zero. Thus, it may be desirable to beable to address both situations when representing, operating upon and/ormanipulating hierarchical sets of data.

For this particular embodiment, a tree with nulls, as described above,may be converted to a tree without nulls. This shall be illustrated, forexample, for nodes labeled with a null, such as for the tree in FIG. 17;however, the claimed subject matter is not limited in scope in thisrespect, of course. In this particular embodiment, the nulls are deletedfrom those nodes having a null. This is illustrated, for example, byFIG. 18.

Referring now to FIG. 19, node values are also deleted for those nodesstoring the value zero. Furthermore, a single new node is attached tothe nodes previously storing the zero values. The edge between the priornodes and the new nodes are labeled with a zero value. For nodes with avalue k that is a power of 2, including the value one, the node value isdeleted and (log₂(k)+2) new nodes are attached, labeling the edgebetween the new nodes and the old nodes with a zero value. This isillustrated, for example, in FIG. 20.

The remaining node values comprise non-powers of two that are 3 orlarger. These node values are factored into one or more non-compositenumerals. For the resulting non-composite numerals, the non-compositenumeral is replaced with the tag value of the index, for example, i, forthe non-composite. For this particular embodiment, the term, tag valueof index, i, for example, refers to a binary edge labeled tree in thepreviously discussed embodiment of a string-tree association thatcorresponds to a binary string associated with the binary numeral for i.The new edges are labeled with a binary value of zero. This isillustrated, for example, in FIG. 21. The remaining edges are labeledwith a binary value of one. This tree is illustrated, for example, inFIG. 22.

Likewise, in an alternative embodiment, a node labeled tree may comprisefixed length tuples of numerals. For such an embodiment, such multiplenumerals may be combined into a single numeral, such as by employingCantor pairing operations, for example. See, for example, Logical NumberTheory, An Introduction, by Craig Smorynski, pp, 14-23, available fromSpringer-Verlag, 1991. This approach should produce a tree to which thepreviously described embodiments may then be applied. Furthermore, forone embodiment, a tree in which nodes are labeled with numerals ornumerical data, rather than binary data, may be converted to a binaryedge labeled tree and/or binary node labeled tree, and, for anotherembodiment, a tree in which edges are labeled with numerals or numericaldata, rather than binary data, may be converted to a binary edge labeledtree and/or binary node labeled tree.

Furthermore, a tree in which both the nodes and the edges are labeledmay be referred to in this context as a feature tree and may beconverted to a binary edge labeled tree and/or binary node labeled tree.For example, without intending to limit the scope of the claimed subjectmatter, in one approach, a feature tree may be converted by convertingany labeled node with its labeled outgoing edge to an ordered pair oflabels for the particular node. Using the embodiment described above,this tree may then be converted to a binary edge labeled tree.

In yet another embodiment, for trees in which data labels do notcomprise simply natural numerals, such as, as one example, trees thatinclude negative numerals, such data labels may be converted to anordered pair of numerals. For example, the first numeral may represent adata type. Examples include a data type such as negative, dollars, etc.As described above, such trees may also be converted to binary edgelabeled trees, such as by applying the previously described embodiment,for example.

A similar conversion exists for binary strings. In this context,therefore, depending upon the particular embodiment, for example, binarystrings may be depicted as binary edge labeled strings, binary nodelabeled strings, and the like. Furthermore, for those embodiments inwhich labels are not binary, for example, a conversion may be made to abinary node labeled string and/or binary edge labeled string, forexample. Alternatively, as described previously in connection withtrees, depending on the particular embodiment, strings in thesealternative forms (e.g., numerals versus binary data) may be manipulatedand/or operated upon directly. Thus, in this context, when referring toan embodiment of an association, the association is meant to refer anassociation between strings and trees, where binary edge labeled stringsand binary edge labeled trees are one particular embodiment. Thus, otherembodiments may provide a similar type of association, however, suchembodiments may alternative use binary node labels and the like. Asimilar proposition applies for association embodiments between naturalnumbers and strings and/or between natural numbers and trees. Thus, whenreferring to an embodiment of an association, the association is meantto refer a particular association between natural numerals and trees orbetween natural numerals and strings, where here binary edge labeledstrings and binary edge labeled trees are one particular embodiment.

As previously described, trees may be employed to graphically representa hierarchy of data or a hierarchy of a set of data. This has beenillustrated in some detail for binary edge labeled trees, for example.As the previous figures, illustrate, however, such graphicalhierarchical representations typically employ two spatial dimensions todepict the relationship among different pieces of data. This may bedisadvantageous in some situations where a one dimensionalrepresentation or arrangement of symbols, such as is employed withalphabetic letters, for example, that are combined to create a linearcollection of successive symbols or notations, such as words, would bemore convenient.

FIG. 10 is a table that illustrates one particular embodiment ofemploying symbols, concatenated along one spatial dimension, here fromleft to right, by convention, to represent such a data hierarchy. Thetable includes four columns. The first column denotes natural numerals.The second column denotes binary strings, such as those are previouslydescribed. The third column denotes a one dimensional arrangement ofsymbols employed to represent the binary edge labeled trees for thatparticular position. Of course, as previously described in connectionwith prior embodiments, the claimed subject matter is not limited inscope to binary strings or binary edge labeled trees. Thus, inalternative embodiments, any tree may be represented with symbolsorganized in a one dimensional arrangement, as is demonstrated with thisparticular embodiment.

For this particular embodiment, the symbols are parsed to reduceredundancy of information. For this particular embodiment, the symbols#₀, and #₁ are employed, to represent zero-push and one-push operations,respectively, although, of course, this is merely an embodiment. Anysymbols may be employed, including, for example, symbols differentiatedby color, symbols differentiated by shape, and the like.

A first symbol, such as for this particular embodiment, a “1,” may beemployed to represent a node or an edge. For this particular embodiment,“1” represents a node. Additional symbols may be employed, in thisparticular embodiment, to represent labels of a node or an edge. Again,in this particular embodiment, for binary edge labeled trees, symbolsare employed to represent labels of edges. Again, for this particularembodiment, two different labels are employed, a label for a binary oneand a different label for a binary zero. Of course, “1” and “0” may beemployed and another symbol may be employed to represent nodes in analternative embodiment. However, using symbols other than “1” and “0” torepresent binary one and binary zero reduces confusion with othertypical and/or well-known binary numeral schemes.

Here, the linear or successive order of the symbols is employed torepresent the graphical hierarchy. Thus, for a labeled edge, the labelfor that particular edge precedes the symbols that represent the nodesfor the particular edge. As one example, consider the representation ofthe binary edge labeled tree associated with position two. Asillustrated by FIG. 10 in comparison with FIG. 8, the label representingbinary zero is provided immediately ahead of two adjacent symbols fornodes. The “1” immediately after the label represents the bottom nodeand the next “1” represents the root node. Thus, the number of “1”srepresents the number of nodes for the particular binary edge labeledtree in this particular embodiment. The particular label thereforeindicates here that the edge connecting the nodes is labeled binaryzero. Likewise, a similar approach is employed for the representation ofthe tree for position three. Thus, the label of the edge connecting thetwo nodes comprises a binary one. In contrast, for the position for thebinary edge labeled tree of position four, the first symbol representsthe binary “0” label and the second symbol represents the bottom nodefor that edge. The other node for that particular edge is the root node.The symbols are then repeated, representing another edge labeled with abinary zero connected to the root node.

This above may be contrasted with the representation for position fiveof this association embodiment in which the labels for the edges areimmediately adjacent each other and symbols for the nodes associatedwith the labeled edges are after the two adjacent label symbols. Thisindicates that the edges are connected to each other, as illustrated bythe graphical representation of this particular binary edge labeledtree, rather than each being connected to the root note, as in the priortree. Of course, the successive edges are then connected to the rootnode at the top, illustrated by the final “1” at position five. Thus,using this particular embodiment, it is possible to representation allbinary edge labeled trees using three symbols going from left to right.

It will, of course, be understood that, although particular embodimentshave just been described, the claimed subject matter is not limited inscope to a particular embodiment or implementation. For example, oneembodiment may be in hardware, such as implemented to operate on adevice or combination of devices, for example, whereas anotherembodiment may be in software. Likewise, an embodiment may beimplemented in firmware, or as any combination of hardware, software,and/or firmware, for example. Likewise, although the claimed subjectmatter is not limited in scope in this respect, one embodiment maycomprise one or more articles, such as a storage medium or storagemedia. This storage media, such as, one or more CD-ROMs and/or disks,for example, may have stored thereon instructions, that when executed bya system, such as a computer system, computing platform, or othersystem, for example, may result in an embodiment of a method inaccordance with the claimed subject matter being executed, such as oneof the embodiments previously described, for example. As one potentialexample, a computing platform may include one or more processing unitsor processors, one or more input/output devices, such as a display, akeyboard and/or a mouse, and/or one or more memories, such as staticrandom access memory, dynamic random access memory, flash memory, and/ora hard drive, although, again, the claimed subject matter is not limitedin scope to this example.

In the preceding description, various aspects of the claimed subjectmatter have been described. For purposes of explanation, specificnumbers, systems and/or configurations were set forth to provide athorough understanding of the claimed subject matter. However, it shouldbe apparent to one skilled in the art having the benefit of thisdisclosure that the claimed subject matter may be practiced without thespecific details. In other instances, well-known features were omittedand/or simplified so as not to obscure the claimed subject matter. Whilecertain features have been illustrated and/or described herein, manymodifications, substitutions, changes and/or equivalents will now occurto those skilled in the art. It is, therefore, to be understood that theappended claims are intended to cover all such modifications and/orchanges as fall within the true spirit of the claimed subject matter.

1. A method of combining at least two trees comprising: converting saidat least two trees to numerals; multiplying said numerals; andconverting the product of said numerals to a tree.
 2. The method ofclaim 1, wherein said trees comprise binary edge labeled trees.
 3. Themethod of claim 1, wherein said at least two trees are converted tobinary edge labeled trees.
 4. The method of claim 1, wherein at leastone of said at least two trees includes at least one null label.
 5. Themethod of claim 4, wherein said at least one of said at least two treesthat includes at least one null label is converted to a tree without anull label.
 6. The method of claim 5, wherein said tree without a nulltype is converted to a binary edge labeled tree.
 7. The method of claim1, wherein said at least two trees are converted to binary node labeledtrees.
 8. The method of claim 1, wherein said numerals comprises binarynumerals.
 9. The method of claim 1, wherein said numerals are convertedto binary numerals.
 10. A method of combining at least two treescomprising: converting said at least two trees to strings; combiningsaid strings; and converting the combination of strings to a tree. 11.The method of claim 10, wherein said trees comprise binary edge labeledtrees.
 12. The method of claim 10, wherein said at least two trees areconverted to binary edge labeled trees.
 13. The method of claim 10,wherein at least one of said at least two trees includes at least onenull label.
 14. The method of claim 13, wherein said at least one ofsaid at least two trees that includes at least one null label isconverted to a tree without a null label.
 15. The method of claim 14,wherein said tree without a null label is converted to a binary edgelabeled tree.
 16. The method of claim 10, wherein said at least twotrees are converted to binary node labeled trees.
 17. The method ofclaim 10, wherein said strings comprise binary strings.
 18. The methodof claim 17, wherein said binary strings comprise binary edge labeledstrings.
 19. The method of claim 10, wherein said strings comprisebinary strings, and wherein combining said binary strings comprises:converting said binary strings to binary numerals, multiplying saidbinary numerals, and converting the product to a binary string.
 20. Amethod of splitting a first tree into at least second and third treescomprising: converting said first tree to a first numeral; factoringsaid first numeral into at least second and third numerals; andconverting said at least second and third numerals to at least secondand third trees.
 21. The method of claim 20, wherein said first treecomprises a binary edge labeled tree.
 22. The method of claim 20,wherein said first tree is converted to a binary edge labeled tree. 23.The method of claim 20, wherein said first tree includes at least onenull label.
 24. The method of claim 23, wherein said first tree isconverted to a tree without a null label.
 25. The method of claim 24,wherein said tree without a null label is converted to a binary edgelabeled tree.
 26. The method of claim 20, wherein said first tree isconverted to a binary node labeled tree.
 27. The method of claim 20,wherein said numerals comprise binary numerals.
 28. The method of claim20, wherein said numerals are converted to binary numerals.
 29. A methodof splitting a first tree into at least second and third treescomprising: converting said first tree to a first string; splitting saidfirst string into at least second and third strings; and converting saidat least second and third strings to said at least second and thirdtrees.
 30. The method of claim 29, wherein said first tree comprises abinary edge labeled tree.
 31. The method of claim 29, wherein said firsttree is converted to a binary edge labeled tree.
 32. The method of claim29, wherein said first tree includes at least one null label.
 33. Themethod of claim 32, wherein said first tree is converted to a treewithout a null label.
 34. The method of claim 33, wherein said treewithout a null label is converted to a binary edge labeled tree.
 35. Themethod of claim 32, wherein said tree without a null label is convertedto a binary node labeled tree.
 36. The method of claim 29, wherein saidfirst tree is converted to a binary node labeled tree.
 37. The method ofclaim 29, wherein said strings comprise binary strings.
 38. The methodof claim 37, wherein said binary strings are converted to numerals. 39.The method of claim 37, wherein said binary strings comprise binary edgelabeled strings.
 40. The method of claim 29, wherein said stringscomprise binary strings, and wherein splitting said first string into atleast second and third strings comprises: converting said first stringsto binary numerals, factoring said binary numerals, and converting thefactors to binary strings.
 41. A method of converting between a stringand a tree comprising: converting said string to a numeral; convertingsaid numeral to said tree.
 42. The method of claim 41, wherein saidstring comprises a binary string.
 43. The method of claim 42, whereinsaid binary string comprises a binary edge labeled string.
 44. Themethod of claim 41, wherein said tree comprises a binary edge labeledtree.
 45. The method of claim 41, wherein said tree comprises a binarynode labeled tree.
 46. The method of claim 45, wherein said binary nodelabeled tree is converted to a numeral node labeled tree.
 47. The methodof claim 46, wherein said numeral node labeled tree is converted to anumeral node labeled tree with at least one null label.
 48. The methodof claim 41, wherein said string is converted to a binary edge labeledstring.
 49. The method of claim 41, wherein said tree is converted to atree with at least one null label.
 50. The method of claim 41, whereinsaid tree comprises a binary edge labeled tree, wherein said binary edgelabeled tree is converted to a numeral edge labeled tree.
 51. The methodof claim 50, wherein said binary edge labeled tree is converted to anumeral edge labeled tree with at least one null label.
 52. The methodof claim 41, wherein said numeral comprises a binary numeral.
 53. Themethod of claim 41, wherein said numeral is converted to a binarynumeral.
 54. A method of converting between a tree and a stringcomprising: converting said tree to a numeral; converting said numeralto said string.
 55. The method of claim 54, wherein said stringcomprises a binary string.
 56. The method of claim 55, wherein saidbinary string comprises a binary edge labeled string.
 57. The method ofclaim 54, wherein said tree comprises a binary edge labeled tree. 58.The method of claim 54, wherein said tree comprises a binary nodelabeled tree.
 59. The method of claim 54, wherein said tree comprises anumeral node labeled tree, said numeral node labeled tree beingconverted to a binary node labeled tree.
 60. The method of claim 54,wherein said tree comprises a numeral node labeled tree with at leastone null label, said numeral labeled tree being converted to a numeralnode labeled tree without a null label.
 61. The method of claim 54,wherein said tree comprises a numeral edge labeled tree, wherein saidnumeral edge labeled tree is converted to a binary edge labeled tree.62. The method of claim 54, wherein said numeral comprises a binarynumeral.
 63. A method of ordering a set of binary strings comprising:preceding longer binary strings with shorter binary strings; andordering binary strings of the same length based at least upon theirassociated binary values when a binary one is added to the left of thebinary string.
 64. The method of claim 63, wherein said binary stringscomprise binary node labeled strings; and wherein ordering binary nodelabeled strings of the same length comprises ordering binary nodelabeled strings based at least upon the associated binary values. 65.The method of claim 63, wherein said binary strings comprise binary edgelabeled strings; and wherein ordering binary edge labeled strings of thesame length comprises ordering binary edge labeled strings based atleast upon the associated binary values.
 66. A method of enumerating aset of trees comprising: positioning a tree at location k, where k isthe product of u and v, such that said tree is formed by a union oftrees in positions u and v; k, u, and v comprising positive naturalnumerals greater than one.
 67. The method of claim 66, wherein saidtrees comprise binary edge labeled trees.
 68. The method of claim 66,wherein said trees comprise binary node labeled trees.
 69. The method ofclaim 66, wherein said trees comprise numeral edge labeled trees, andfurther comprising: converting said numeral edge labeled trees to binaryedge labeled trees.
 70. The method of claim 66, wherein said treescomprise numeral node labeled trees, and further comprising: convertingsaid numeral node labeled trees to binary node labeled trees.
 71. Themethod of claim 66, and further comprising: positioning a tree atlocation j, where j comprises a non-composite numeral greater than one,such that said tree is formed from a prior tree by attaching a node as aroot node.
 72. The method of claim 71, wherein attaching a node as aroot node further comprises labeling the edge between said node and saidroot node.
 73. The method of claim 72, wherein said trees comprisebinary edge labeled trees, and said edge is labeled either 0 or
 1. 74.The method of claim 73, wherein said edge is labeled with a 0, j beingP(2m−1), where P(2m−1) is the (2m−1)th non-composite numeral of theKeene enumeration of non-composite numerals on positive naturalnumerals.
 75. The method of claim 73, wherein said edge is labeled witha 1, j being P(2m), where P(2m) is the (2m)th non-composite numeral ofthe Keene enumeration of non-composite numerals on positive naturalnumerals.
 76. The method of claim 66, wherein said trees comprisenumeral edge labeled trees, and further comprising: converting saidnumeral edge labeled trees to binary edge labeled trees.
 77. The methodof claim 66, and further comprising: positioning at location k a stringsuch that said string corresponds to a binary value, said binary valuecorresponding to k with the left most binary digit omitted.
 78. Themethod of claim 77, wherein said string comprises a binary string. 79.The method of claim 78, wherein said binary string comprises a binaryedge labeled string.
 80. The method of claim 78, wherein said binarystrings comprises a binary node labeled strings.
 81. A method of formingtrees comprising: forming at least one of a zero-push and a one-push ofa tree.
 82. The method of claim 81, wherein said tree comprises binaryedge labeled tree.
 83. The method of claim 81, wherein forming treesalso comprises forming at least an N-push of said tree, where N is apositive natural numeral greater than one.
 84. The method of claim 81,wherein said tree is converted to a binary edge labeled tree from abinary node labeled tree.
 85. The method of claim 81, wherein said treecomprises a numeral edge labeled tree.
 86. The method of claim 85,wherein said tree is converted to a binary edge labeled tree from saidnumeral edge labeled tree.
 87. The method of claim 85, wherein saidnumeral edge labeled tree includes at least one null label.
 88. Themethod of claim 81, wherein said tree comprises a numeral node labeledtree.
 89. The method of claim 88, wherein said tree is converted to abinary edge labeled tree from said numeral node labeled tree.
 90. Themethod of claim 88, wherein said numerical node labeled tree includes atleast one null label.
 91. The method of claim 81, and furthercomprising: positioning said zero-push tree at position P(2k−1), if k isthe position of said tree in a tree enumeration, P(2k−1) comprising the(2k−1)th non-composite of the Keene enumeration of non-compositenumerals on the positive natural numerals.
 92. The method of claim 81,and further comprising: positioning said one-push tree at positionP(2k), if k is the position of said tree in a tree enumeration, P(2k)comprising the (2k)th non-composite of the Keene enumeration ofnon-composite numerals on the positive natural numerals.
 93. A method ofrepresenting a graphical hierarchy of data using a one-dimensionalarrangement of symbols comprising: employing a first symbol to representa node or an edge; employing additional symbols to represent labels ofsaid node or edge; wherein the successive order of said symbols providessaid graphical hierarchy.
 94. The method of claim 93, wherein said firstsymbol represents a node; said additional symbols comprising two symbolsthat respectively represent labels 0 and 1; and wherein said graphicalhierarchy comprises a binary edge labeled tree.
 95. The method of claim94, wherein an edge label symbol is provided prior to the symbol for anode associated with the edge that it labels.
 96. The method of claim95, wherein an edge label symbol is provided immediately prior to thesymbol for a node associated with the edge that it labels
 97. The methodof claim 93, wherein said first symbol represents an edge; saidadditional symbols comprising two symbols to respectively represent 0and 1; and wherein said graphical hierarchy comprises a binary nodelabeled tree.
 98. The method of claim 97, wherein a node label symbol isprovided prior to the symbol for an edge associated with the node thatit labels.
 99. The method of claim 98, wherein a node label symbol isprovided immediately prior to the symbol for an edge associated with thenode that it labels.
 100. The method of claim 93, wherein adjacent labelsymbols represent successively connected nodes or edges.
 101. The methodof claim 93, wherein non-adjacent label symbols represent nodes or edgesthat are not successively connected.
 102. An article comprising: astorage medium having stored thereon an association between strings andtree; said association comprising a set of trees and strings,respectively having been enumerated by: positioning a tree at locationk, where k is the product of u and v, such that said tree is formed by aunion of trees in positions u and v; k, u, and v comprising positivenatural numerals greater than one.
 103. The article of claim 102,wherein said trees comprise binary edge labeled trees.
 104. The articleof claim 102, wherein said trees comprise binary node labeled trees.105. The article of claim 102, wherein said trees comprise numeral edgelabeled trees, said numeral edge labeled trees having been converted tobinary edge labeled trees.
 106. The article of claim 102, wherein saidtrees comprise numeral node labeled trees, said numeral node labeledtrees having been converted to binary node labeled trees.
 107. Thearticle of claim 102, wherein said association having been enumeratedby: positioning a tree at location k, where k comprises a non-compositenatural numeral greater than one, such that said tree is formed from aprior tree by attaching a node as a root node.
 108. The article of claim102, wherein said association having been enumerated by: labeling theedge between said node and said root node.
 109. The article of claim108, wherein said trees comprise binary edge labeled trees, and whereinsaid association having been enumerated by: labeling said edge either 0or
 1. 110. The article of claim 109, wherein said association havingbeen enumerated by: said edge being labeled with a
 0. 111. The articleof claim 109, wherein said association having been enumerated by: saidedge being labeled with a
 1. 112. The article of claim 102, wherein saidassociation having been enumerated by: positioning at location k astring such that said string corresponding to a binary value, saidbinary value corresponds to k with the left most binary digit omitted.113. The article of claim 112, wherein said string comprises a binarystring.
 114. The article of claim 113, wherein said binary stringcomprises a binary edge labeled string.
 115. The article of claim 113,wherein said binary string comprises a binary node labeled string. 116.A method of enumerating a set of strings comprising: positioning atlocation k a string such that said string corresponds to a binary value,said binary value corresponding to k with the left most binary digitomitted; k comprising a positive natural number greater than one. 117.The method of claim 116, wherein said strings comprise binary strings.118. The method of claim 117, wherein said binary strings comprisebinary edge labeled strings.
 119. The method of claim 117, wherein saidbinary strings comprise binary node labeled strings.
 120. A method ofordering a set of trees comprising: converting said set of trees tonumerals; ordering said numerals; and converting said numerals to saidset of trees.
 121. The method of claim 120, wherein said trees comprisebinary edge labeled trees.
 122. The method of claim 120, wherein saidtrees comprise binary node labeled trees.
 123. The method of claim 120,wherein said trees comprise numeral edge labeled trees, and furthercomprising: converting said numeral edge labeled trees to binary edgelabeled trees.
 124. The method of claim 120, wherein said trees comprisenumeral node labeled trees, and further comprising: converting saidnumeral node labeled trees to binary node labeled trees.
 125. The methodof claim 120, and further comprising ordering a set of stringscomprising: preceding longer binary strings with shorter binary strings;and ordering binary strings of the same length based upon theirassociated binary values when a binary one is added to the left of thebinary string; wherein a relationship exists between the ordered treesand the ordered binary strings
 126. The method of claim 125, whereinsaid binary strings comprise binary node labeled strings; and whereinordering binary node labeled strings of the same length comprisesordering binary node labeled strings based at least in part upon theassociated binary values.
 127. The method of claim 125, wherein saidbinary strings comprise binary edge labeled strings; and whereinordering binary edge labeled strings of the same length comprisesordering binary edge labeled strings based at least in part upon theassociated binary values.
 128. An article comprising: a storage mediumhaving stored thereon an association between a set of strings and a setof trees; said set of trees having been ordered by: converting said setof trees to numerals; ordering said numerals; and converting saidnumerals to said set of trees.
 129. The article of claim 128, whereinsaid trees comprise binary edge labeled trees.
 130. The article of claim128, wherein said trees comprise binary node labeled trees.
 131. Thearticle of claim 128, wherein said trees comprise numeral edge labeledtrees, said numeral edge labeled trees having been converted to binaryedge labeled trees.
 132. The article of claim 128, wherein said treescomprise numeral node labeled trees, said numeral node labeled treeshaving been converted to binary node labeled trees.
 133. The article ofclaim 128, and further comprising ordering a set of strings comprising:preceding longer binary strings with shorter binary strings; andordering binary strings of the same length based upon their associatedbinary values when a binary one is added to the left of the binarystring; wherein a relationship exists between the ordered trees and theordered binary strings.
 134. The article of claim 133, wherein saidbinary strings comprise binary node labeled strings; and whereinordering binary node labeled strings of the same length comprisesordering binary node labeled strings based upon the associated binaryvalues.
 135. The article of claim 133, wherein said binary stringscomprise binary edge labeled strings; and wherein ordering binary edgelabeled strings of the same length comprises ordering binary edgelabeled strings based upon the associated binary values.
 136. Anapparatus comprising: a computing platform; said computing platformbeing adapted to convert at least two trees to numerals, multiply saidnumerals, and convert the product of said numerals to a tree.
 137. Theapparatus of claim 136, wherein said trees comprise binary edge labeledtrees.
 138. The apparatus of claim 136, wherein said computing platformis adapted to convert said at least two trees to binary edge labeledtrees.
 139. The apparatus of claim 136, wherein at least one of said atleast two trees includes at least one null label.
 140. The apparatus ofclaim 139, wherein said computing platform is adapted to convert said atleast one of said at least two trees that includes at least one nulllabel to a tree without a null label.
 141. The apparatus of claim 140,wherein said computing platform is adapted to convert said tree withouta null type to a binary edge labeled tree.
 142. The apparatus of claim136, wherein said computing platform is adapted to convert said at leasttwo trees to binary node labeled trees.
 143. The apparatus of claim 136,wherein said numerals comprise binary numerals.
 144. The apparatus ofclaim 136, wherein said computing platform is adapted to convert saidnumerals to binary numerals.
 145. An apparatus comprising: a computingplatform; said computing platform being adapted to convert at least twotrees to strings, combine said strings, and convert the combination ofstrings to a tree.
 146. The apparatus of claim 145, wherein said treescomprise binary edge labeled trees.
 147. The apparatus of claim 145,wherein said computing platform is adapted to convert said at least twotrees to binary edge labeled trees.
 148. The apparatus of claim 145,wherein at least one of said at least two trees includes at least onenull label.
 149. The apparatus of claim 148, wherein said computingplatform is adapted to convert said at least one of said at least twotrees that includes at least one null label to a tree without a nulllabel.
 150. The apparatus of claim 149, wherein said computing platformis adapted to convert said tree without a null label to a binary edgelabeled tree.
 151. The apparatus of claim 145, wherein said computingplatform is adapted to convert said at least two trees to binary nodelabeled trees.
 152. The apparatus of claim 145, wherein said stringscomprise binary strings.
 153. The apparatus of claim 152, wherein saidbinary strings comprise binary edge labeled strings.
 154. The apparatusof claim 145, wherein said strings comprise binary strings, and whereinsaid computing platform is adapted to combine said binary strings byconverting said binary strings to binary numerals, multiplying saidbinary numerals, and converting the product to a binary string.
 155. Anapparatus comprising: a computing platform; said computing platformbeing adapted to convert a first tree to a first numeral, factor saidfirst numeral into at least second and third numerals, and convert saidsecond and third numerals to second and third trees.
 156. The apparatusof claim 155, wherein said first tree comprises a binary edge labeledtree.
 157. The apparatus of claim 155, wherein said computing platformis adapted to convert said first tree to a binary edge labeled tree.158. The apparatus of claim 155, wherein said first tree includes atleast one null label.
 159. The apparatus of claim 158, wherein saidcomputing platform is adapted to convert said first tree to a treewithout a null label.
 160. The apparatus of claim 155, wherein saidcomputing platform is adapted to convert said first tree to a binarynode labeled tree.
 161. The apparatus of claim 155, wherein saidnumerals comprise binary numerals.
 162. The apparatus of claim 155,wherein said computing platform comprises a computer.
 163. The apparatusof claim 155, wherein said computing platform is adapted to convert saidnumerals to binary numerals.
 164. An apparatus comprising: a computingplatform; said computing platform being adapted to convert a first treeto a first string, split said first string into at least second andthird strings, and convert said second and third strings to second andthird trees.
 165. The apparatus of claim 164, wherein said first treecomprises a binary edge labeled tree.
 166. The apparatus of claim 165,wherein said computing platform is adapted to convert said first tree toa binary edge labeled tree.
 167. The apparatus of claim 164, whereinsaid first tree includes at least one null label.
 168. The apparatus ofclaim 166, wherein said computing platform is adapted to convert saidfirst tree to a tree without a null label.
 169. The apparatus of claim168, wherein said computing platform is adapted to convert said treewithout a null label to a binary edge labeled tree.
 170. The apparatusof claim 167, wherein said computing platform is adapted to convert saidtree without a null label to a binary node labeled tree.
 171. Theapparatus of claim 164, wherein said computing platform is adapted toconvert said first tree to a binary node labeled tree.
 172. Theapparatus of claim 164, wherein said strings comprise binary strings.173. The apparatus of claim 172, wherein said computing platform isadapted to convert said binary strings to numerals.
 174. The apparatusof claim 172, wherein said binary strings comprise binary edge labeledstrings.
 175. The apparatus of claim 164, wherein said strings comprisebinary strings, and wherein said computing platform is adapted to splitsaid first string to at least second and third strings by: convertingsaid first string to a binary numeral, factoring said binary numeral,and converting the factors to binary strings.
 176. An apparatuscomprising: a computing platform; said computing platform being adaptedto convert a string to a numeral, and convert said numeral to a tree.177. The apparatus of claim 176, wherein said string comprises a binarystring.
 178. The apparatus of claim 177, wherein said binary stringcomprises a binary edge labeled string.
 179. The apparatus of claim 176,wherein said tree comprises a binary edge labeled tree.
 180. Theapparatus of claim 176, wherein said tree comprises a binary nodelabeled tree.
 181. The apparatus of claim 180, wherein said computingplatform is adapted to convert said binary node labeled to a numeralnode labeled tree.
 182. The apparatus of claim 181, wherein saidcomputing platform is adapted to convert said numerical node labeledtree to a numeral node labeled tree with at least one null label. 183.The apparatus of claim 176, wherein said computing platform is adaptedto convert said string to a binary edge labeled string.
 184. Theapparatus of claim 176, wherein said computing platform is adapted toconvert said tree to a tree with at least one null label.
 185. Theapparatus of claim 176, wherein said tree comprises a binary edgelabeled tree, wherein said computing platform is adapted to convert saidbinary edge labeled tree to a numeral edge labeled tree.
 186. Theapparatus of claim 185, wherein said computing platform is adapted toconvert said binary edge labeled tree to a numeral edge labeled treewith at least one null label.
 187. The apparatus of claim 176, whereinsaid numeral comprises a binary numeral.
 188. The apparatus of claim176, wherein said computing platform is adapted to convert said numeralto a binary numeral.
 189. An apparatus comprising: a computing platform;said computing platform being adapted to convert a tree to a numeral,and convert said numeral to a string.
 190. The apparatus of claim 189,wherein said string comprises a binary string.
 191. The apparatus ofclaim 190, wherein said binary string comprises a binary edge labeledstring.
 192. The apparatus of claim 189, wherein said tree comprises abinary edge labeled tree.
 193. The apparatus of claim 189, wherein saidtree comprises a binary node labeled tree.
 194. The apparatus of claim189, wherein said tree comprises a numeral node labeled tree, whereinsaid computing platform is adapted to convert said numeral node labeledtree to a binary node labeled tree.
 195. The apparatus of claim 189,wherein said tree comprises a numeral node labeled tree with at leastone null label, wherein said computing platform is adapted to convertsaid numeral labeled tree to a numeral node labeled tree without a nulllabel.
 196. The apparatus of claim 189, wherein said tree comprises anumeral edge labeled tree, wherein said computing platform is adapted toconvert said numeral edge labeled tree to a binary edge labeled tree.197. The apparatus of claim 189, wherein said numeral comprises a binarynumeral.
 198. An apparatus comprising: a computing platform; saidcomputing platform being adapted to order binary strings by: precedinglonger binary strings with shorter binary strings; and ordering binarystrings of the same length based at least upon their associated binaryvalues when a binary one is added to the left of the binary string. 199.The apparatus of claim 198, wherein said binary strings comprise binaryedge labeled strings; and wherein said computing platform is adapted toorder binary edge labeled strings of the same length based at least uponthe associated binary values.
 200. The apparatus of claim 198, whereinsaid binary strings comprise binary node labeled strings; and whereinsaid computing platform is adapted to order binary node labeled stringsof the same length based at least upon the associated binary values.201. An apparatus comprising: a computing platform; said platform beingadapted to enumerate trees by: positioning a tree at location k, where kis the product of u and v, such that said tree is formed by a union oftrees in positions u and v; k, u, and v comprising positive naturalnumerals greater than one.
 202. The apparatus of claim 201, wherein saidtrees comprise binary edge labeled trees.
 203. The apparatus of claim201, wherein said trees comprise binary node labeled trees.
 204. Theapparatus of claim 201, wherein said trees comprise numeral edge labeledtrees, and wherein said computing platform is adapted to convert saidnumeral edge labeled trees to binary edge labeled trees.
 205. Theapparatus of claim 201, wherein said trees comprise numeral node labeledtrees, and wherein said computing platform is adapted to convert saidnumeral node labeled trees to binary node labeled trees.
 206. Theapparatus of claim 201, wherein said computing platform is adapted toposition a tree at location j, where j comprises a non-composite numeralgreater than one, such that said tree is formed from a prior tree byattaching a node as a root node.
 207. The apparatus of claim 206,wherein computing platform being adapted to label the edge between saidnode and said root node.
 208. The apparatus of claim 207, wherein saidtrees comprise binary edge labeled trees, and wherein said computingplatform is adapted to label said edge either 0 or
 1. 209. The apparatusof claim 208, wherein said computing platform is adapted to label saidedge with a 0, j being P(2m−1), where P(2m−1) is the (2m−1)thnon-composite numeral of the Keene enumeration of non-composite numeralson positive natural numerals.
 210. The apparatus of claim 208, whereinsaid computing platform is adapted to label said edge with a 1, j beingP(2m), where P(2m) is the (2m)th non-composite numeral of the Keeneenumeration of non-composite numerals on positive natural numerals. 211.The apparatus of claim 201, wherein said trees comprise numeral edgelabeled trees, and wherein said computing platform is adapted to convertsaid numeral edge labeled trees to binary edge labeled trees.
 212. Theapparatus of claim 201, wherein said computing platform is adapted toposition at location k a string such that said string corresponds to abinary value, said binary value corresponding to k with the left mostbinary digit omitted.
 213. The apparatus of claim 212, wherein saidstrings comprise binary strings.
 214. The apparatus of claim 213,wherein said binary strings comprise binary edge labeled strings. 215.The apparatus of claim 213, wherein said binary strings comprise binarynode labeled strings:
 216. An apparatus comprising: a computingplatform; said computing platform being adapted to form at least one ofa zero-push and a one-push of a tree.
 217. The apparatus of claim 216,wherein said tree comprises a binary edge labeled tree.
 218. Theapparatus of claim 216, wherein said computing platform is adapted toform a tree comprising at least an N-push of said tree, where N is apositive natural numeral greater than one.
 219. The apparatus of claim216, wherein said computing platform is adapted to convert said tree toa binary edge labeled tree.
 220. The apparatus of claim 219, whereinsaid computing platform is adapted to convert said tree to said binaryedge labeled tree from a binary node labeled tree.
 221. The apparatus ofclaim 216, wherein said tree comprises a numeral edge labeled tree. 222.The apparatus of claim 221, wherein said computing platform is adaptedto convert said tree to a binary edge labeled tree from said numeraledge labeled tree.
 223. The apparatus of claim 221, wherein said numeraledge labeled tree includes at least one null label.
 224. The apparatusof claim 216, wherein said tree comprises a numeral node labeled tree.225. The apparatus of claim 224, wherein said computing platform isadapted to convert said tree to a binary edge labeled tree from saidnumeral node labeled tree.
 226. The apparatus of claim 224, wherein saidnumerical node labeled tree includes at least one null label.
 227. Anapparatus comprising: a computing platform; said computing platformbeing adapted to employ a first symbol to represent a node or an edge,and employ additional symbols to represent labels of said node or edge,wherein the successive order of said symbols provides a graphicalhierarchy.
 228. The apparatus of claim 227, wherein said first symbolrepresents a node; said additional symbols comprise two symbols thatrespectively represent labels 0 and 1; and wherein said graphicalhierarchy comprises a binary edge labeled tree.
 229. The apparatus ofclaim 228, wherein said computing platform is adapted to provide an edgelabel symbol prior to the symbol for a node associated with the edgethat it labels.
 230. The apparatus of claim 229, wherein said computingplatform is adapted to provide an edge label symbol immediately prior tothe symbol for a node associated with the edge that it labels
 231. Theapparatus of claim 227, wherein said first symbol represents an edge;said additional symbols comprise two symbols to respectively represent 0and 1; and wherein said graphical hierarchy comprises a binary nodelabeled tree.
 232. The apparatus of claim 231, wherein said computingplatform is adapted to provide a node label symbol prior to the symbolfor an edge associated with the node that it labels.
 233. The apparatusof claim 232, wherein said computing platform is adapted to provide anode label symbol immediately prior to the symbol for an edge associatedwith the node that it labels.
 234. The apparatus of claim 227, whereinsaid computing platform is adapted to provide adjacent label symbols torepresent successively connected nodes or edges.
 235. The apparatus ofclaim 227, wherein said computing platform is adapted to providenon-adjacent label symbols to represent nodes or edges that are notsuccessively connected.
 236. An apparatus comprising: a computingplatform; said computing platform being adapted to position at locationk a string such that said string corresponds to a binary value, saidbinary value corresponding to k with the left most binary digit omitted.237. The apparatus of claim 236, wherein said strings comprise binarystrings.
 238. The apparatus of claim 237, wherein said binary stringscomprise binary edge labeled strings.
 239. The apparatus of claim 237,wherein said binary strings comprise binary node labeled string.
 240. Anapparatus comprising: a computing platform; said computing platformbeing adapted to convert a set of trees to numerals, order saidnumerals, and convert said numerals to said set of trees.
 241. Theapparatus of claim 240, wherein said trees comprise binary edge labeledtrees.
 242. The apparatus of claim 240, wherein said trees comprisebinary node labeled trees.
 243. The apparatus of claim 240, wherein saidtrees comprise numeral edge labeled trees, and wherein said computingplatform is adapted to convert said numeral edge labeled trees to binaryedge labeled trees.
 244. The apparatus of claim 240, wherein said treescomprise numeral node labeled trees, and wherein said computing platformis adapted to convert said numeral node labeled trees to binary nodelabeled trees.
 245. The apparatus of claim 240, wherein said computingplatform is adapted to order a set of binary strings by: precedinglonger binary strings with shorter binary strings; and ordering binarystrings of the same length based upon their associated binary valueswhen a binary one is added to the left of the binary string; wherein arelationship exists between the ordered trees and the ordered binarystrings.
 246. The apparatus of claim 245, wherein said binary stringscomprise binary node labeled strings; and wherein said computingplatform is adapted to order binary node labeled strings of the samelength based at least in part upon the associated binary values. 247.The apparatus of claim 245, wherein said binary strings comprise binaryedge labeled strings; and wherein said computing platform is adapted toorder binary edge labeled strings of the same length based at least inpart upon the associated binary values.
 248. A set of hierarchical datacomprising: a tree having been formed by: converting at least two othertrees to numerals; multiplying said numerals; and converting the productof said numerals to said tree.
 249. The set of hierarchical data ofclaim 248, wherein said trees comprise binary edge labeled trees. 250.The set of hierarchical data of claim 248, wherein said tree having beenformed by: said at least two other trees being converted to binary edgelabeled trees.
 251. The set of hierarchical data of claim 248, whereintree having been formed by: at least one of said at least two othertrees including at least one null label.
 252. The set of hierarchicaldata of claim 251, wherein said tree having been formed by: said atleast one of said at least two other trees that includes at least onenull label being converted to a tree without a null label.
 253. The setof hierarchical data of claim 252, wherein said tree having been formedby: said tree without a null type being converted to a binary edgelabeled tree.
 254. The set of hierarchical data of claim 248, whereinsaid tree having been formed by: said at least two other trees beingconverted to binary node labeled trees.
 255. The set of hierarchicaldata of claim 248, wherein said numerals comprise binary numerals. 256.The set of hierarchical data of claim 248, wherein said numerals areconverted to binary numerals.
 257. A set of hierarchical datacomprising: a tree having been formed by: converting at least two othertrees to strings; combining said strings; and converting the combinationof strings to said tree.
 258. The set of hierarchical data of claim 257,wherein said trees comprise binary edge labeled trees.
 259. The set ofhierarchical data of claim 257, wherein said tree having been formed by:said at least two other trees being converted to binary edge labeledtrees.
 260. The set of hierarchical data of claim 257, wherein said treehaving been formed by: at least one of said at least two other treesincluding at least one null label.
 261. The set of hierarchical data ofclaim 260, wherein said tree having been formed by: said at least one ofsaid at least two other trees that includes at least one null labelbeing converted to a tree without a null label.
 262. The set ofhierarchical data of claim 261, wherein said tree having been formed by:said tree without a null label being converted to a binary edge labeledtree.
 263. The set of hierarchical data of claim 257, wherein said treehaving been formed by: said at least two other trees being converted tobinary node labeled trees.
 264. The set of hierarchical data of claim257, wherein said strings comprise binary strings.
 265. The set ofhierarchical data of claim 264, wherein said binary strings comprisebinary edge labeled strings.
 266. The set of hierarchical data of claim257, wherein said strings comprise binary strings, and wherein saidcombined binary strings are formed by: converting said binary strings tobinary numerals, multiplying said binary numerals, and converting theproduct to a binary string.
 267. A set of hierarchical data comprising:a first and second tree having been formed by: converting a third treeto a first numeral; factoring said first numeral into at least secondand third numerals; and converting said second and third numerals tosaid first and second trees.
 268. The set of hierarchical data of claim267, wherein said third tree comprises a binary edge labeled tree. 269.The set of hierarchical data of claim 267, wherein said first and secondtrees having been formed by: said third tree being converted to a binaryedge labeled tree.
 270. The set of hierarchical data of claim 267,wherein said first and second trees having been formed by: said thirdtree including at least one null label.
 271. The set of hierarchicaldata of claim 270, wherein said first and second trees having beenformed by: said third tree being converted to a tree without a nulllabel.
 272. The set of hierarchical data of claim 271, wherein saidfirst and second trees having been formed by: said tree without a nulllabel being converted to a binary edge labeled tree.
 273. The set ofhierarchical data of claim 267, wherein said first and second treeshaving been formed by: said third tree being converted to a binary nodelabeled tree.
 274. The set of hierarchical data of claim 267, whereinsaid numerals comprise binary numerals.
 275. The set of hierarchicaldata of claim 267, wherein said first and second trees having beenformed by: said numerals being converted to binary numerals.
 276. A setof hierarchical data comprising: a first and second tree having beenformed by: converting a third tree to a first string; splitting saidfirst string into at least second and third strings; converting saidsecond and third strings to said first and second trees.
 277. The set ofhierarchical data of claim 276, wherein said third tree comprises abinary edge labeled tree.
 278. The set of hierarchical data of claim276, wherein said first and second trees having been formed by: saidthird tree being converted to a binary edge labeled tree.
 279. The setof hierarchical data of claim 276, wherein s said first and second treeshaving been formed by: said third tree including at least one nulllabel.
 280. The set of hierarchical data of claim 279, wherein saidfirst and second trees having been formed by: said third tree beingconverted to a tree without a null label.
 281. The set of hierarchicaldata of claim 280, wherein said first and second trees having beenformed by: said tree without a null label being converted to a binaryedge labeled tree.
 282. The set of hierarchical data of claim 280,wherein said first and second trees having been formed by: said treewithout a null label being converted to a binary node labeled tree. 283.The set of hierarchical data of claim 276, wherein said first and secondtrees having been formed by: said third tree being converted to a binarynode labeled tree.
 284. The set of hierarchical data of claim 276,wherein said strings comprise binary strings.
 285. The set ofhierarchical data of claim 284, wherein said first and second treeshaving been formed by: said binary strings being converted to numerals.286. The set of hierarchical data of claim 284, wherein said binarystrings comprise binary edge labeled strings.
 287. The set ofhierarchical data of claim 276, wherein said strings comprise binarystrings, and wherein said second and third string are formed by:converting said first string to a binary numeral, factoring said binarynumeral, and converting the factors to said second and third binarystrings.
 288. A set of hierarchical data comprising: a tree having beenformed by: converting a string to a numeral; and converting said numeralto said tree.
 289. The set of hierarchical data of claim 288, whereinsaid string comprises a binary string.
 290. The set of hierarchical dataof claim 289, wherein said binary string comprises a binary edge labeledstring.
 291. The set of hierarchical data of claim 288, wherein saidtree comprises a binary edge labeled tree.
 292. The set of hierarchicaldata of claim 288, wherein said tree comprises a binary node labeledtree.
 293. The set of hierarchical data of claim 292, wherein said treehaving been formed by: said binary node labeled tree being converted toa numeral node labeled tree.
 294. The set of hierarchical data of claim293, wherein said tree having been formed by: said numerical nodelabeled tree being converted to a numeral node labeled tree with atleast one null label.
 295. The set of hierarchical data of claim 288,wherein said tree having been formed by: said string being converted toa binary edge labeled string.
 296. The set of hierarchical data of claim288, wherein said tree having been formed by: said tree being convertedto a tree with at least one null label.
 297. The set of hierarchicaldata of claim 288, wherein said tree comprises a binary edge labeledtree, wherein said tree having been formed by: said binary edge labeledtree being converted to a numeral edge labeled tree.
 298. The set ofhierarchical data of claim 297, wherein said tree having been formed by:said binary edge labeled tree being converted to a numeral edge labeledtree with at least one null label.
 299. The set of hierarchical data ofclaim 288, wherein said numeral comprises a binary numeral.
 300. The setof hierarchical data of claim 288, wherein said tree having been formedby: said numeral being converted to a binary numeral.
 301. A set ofhierarchical data comprising: a string having been formed by: convertinga tree to a numeral; and converting said numeral to said string. 302.The set of hierarchical data of claim 301, wherein said string comprisesa binary string.
 303. The set of hierarchical data method of claim 302,wherein said binary string comprises a binary edge labeled string. 304.The set of hierarchical data of claim 301, wherein said tree comprises abinary edge labeled tree.
 305. The set of hierarchical data of claim301, wherein said tree comprises a binary node labeled tree.
 306. Theset of hierarchical data of claim 301, wherein said tree comprises anumeral node labeled tree, said string having been formed by: saidnumeral node labeled tree being converted to a binary node labeled tree.307. The set of hierarchical data of claim 301, wherein said treecomprises a numeral node labeled tree with at least one null label, saidstring having been formed by: said numeral labeled tree being convertedto a numeral node labeled tree without a null label.
 308. The set ofhierarchical data of claim 301, wherein said tree comprises a numeraledge labeled tree, wherein said string having been formed by: saidnumeral edge labeled tree being converted to a binary edge labeled tree.309. The set of hierarchical data of claim 301, wherein said numeralcomprises a binary numeral.
 310. An ordered set of data comprising: aset of strings having been ordered by: preceding longer binary stringswith shorter binary strings; and ordering binary strings of the samelength based at least upon their associated binary values when a binaryone is added to the left of the binary string.
 311. The ordered set ofdata of claim 310, wherein said binary strings comprise binary nodelabeled strings; and wherein ordering binary node labeled strings of thesame length comprises ordering binary node labeled strings based atleast upon the associated binary values.
 312. The ordered set of data ofclaim 311, wherein said binary strings comprise binary edge labeledstrings; and wherein ordering binary edge labeled strings of the samelength comprises ordering binary edge labeled strings based at leastupon the associated binary values.
 313. A set of hierarchical datacomprising: a tree in an enumeration of trees having been formed by:positioning said tree at a location k, where k is the product of u andv, such that said tree is formed by a union of trees in positions u andv; k, u, and v comprising positive natural numerals greater than one.314. The set of hierarchical data of claim 313, wherein said treescomprise binary edge labeled trees.
 315. The set of hierarchical data ofclaim 313, wherein said trees comprise binary node labeled trees. 316.The set of hierarchical data of claim 313, wherein said trees comprisenumeral edge labeled trees, said tree in said enumeration of treeshaving been formed by: converting said numeral edge labeled trees tobinary edge labeled trees.
 317. The set of hierarchical data of claim313, wherein said trees comprise numeral node labeled trees, said treein said enumeration of trees having been formed by: converting saidnumeral node labeled trees to binary node labeled trees.
 318. The set ofhierarchical data of claim 313, said tree in said enumeration of treeshaving been formed by: positioning a tree at location j, where jcomprises a non-composite numeral greater than one, such that said treeis formed from a prior tree by attaching a node as a root node.
 319. Theset of hierarchical data of claim 318, wherein said tree in saidenumeration of trees having been formed by: labeling the edge betweensaid node and said root node.
 320. The set of hierarchical data of claim319, wherein said trees comprise binary edge labeled trees, and saidtree in said enumeration of trees having been formed by: said edge beinglabeled either 0 or
 1. 321. The set of hierarchical data of claim 320,said tree in said enumeration of trees having been formed by: said edgebeing labeled with a 0, j being P(2m−1), where P(2m−1) is the (2m−1)thnon-composite of the Keene enumeration of non-composite numerals on thepositive natural numerals.
 322. The set of hierarchical data of claim320, said tree in said enumeration of trees having been formed by: saidedge being labeled with a 1, j being P(2m), where P(2m) is the (2m)thnon-composite of the Keene enumeration of non-composite numerals on thepositive natural numerals.
 323. The set of hierarchical data of claim313, wherein said trees comprise numeral edge labeled trees, and saidtree in said enumeration of trees having been formed by: converting saidnumeral edge labeled trees to binary edge labeled trees.
 324. The set ofhierarchical data of claim 313, and further comprising: a stringpositioned at location k, said string having been formed based at leastin part on a binary value corresponding to k with the left most binarydigit omitted.
 325. The set of hierarchical data of claim 324, whereinsaid string comprises a binary string.
 326. The set of hierarchical dataof claim 325, wherein said binary string comprises a binary edge labeledstring.
 327. The set of hierarchical data of claim 325, wherein saidbinary string comprises a binary node labeled string.
 328. Ahierarchical set of data comprising: a tree having been formed by atleast one of a zero-push and a one-push of another tree.
 329. Thehierarchical set of data of claim 328, wherein said another treecomprises a binary edge labeled tree.
 330. The hierarchical set of dataof claim 328, and further comprising: yet another tree having beenformed by at least an N-push of said another tree, where N is a positivenatural numeral greater than one.
 331. The hierarchical set of data ofclaim 328, wherein said tree having been formed being converted to abinary edge labeled tree.
 332. The hierarchical set of data of claim331, wherein said tree having been formed being converted to a binaryedge labeled tree from a binary node labeled tree.
 333. The hierarchicalset of data of claim 328, wherein said tree comprises a numeral edgelabeled tree.
 334. The hierarchical set of data of claim 333, whereinsaid tree having been formed being converted to a binary edge labeledtree from said numeral edge labeled tree.
 335. The hierarchical set ofdata of claim 333, wherein said numeral edge labeled tree including atleast one null label.
 336. The hierarchical set of data of claim 328,wherein said tree comprises a numeral node labeled tree.
 337. Thehierarchical set of data of claim 336, wherein said tree having beenformed being converted to a binary edge labeled tree from said numeralnode labeled tree.
 338. The hierarchical set of data of claim 336,wherein said numerical node labeled tree including at least one nulllabel.
 339. The hierarchical set of data of claim 328, wherein said treebeing located at position P(2k−1) in a tree enumeration, said treehaving been formed by said zero-push, where P(x) is the Keeneenumeration of non-composite numerals on the positive natural numerals,k comprising a natural numeral greater than one.
 340. The hierarchicalset of data of claim 328, wherein said tree being located at positionP(2k) in a tree enumeration, said tree having been formed by saidone-push, where P(x) is the Keene enumeration of non-composite numeralson the positive natural numerals, k comprising a natural numeral greaterthan one.
 341. A graphical hierarchy representation comprising: a linearcollection of successively adjacent symbols formed by: employing a firstsymbol to represent a node or an edge; employing additional symbols torepresent labels of said node or edge; wherein the successive order ofsaid symbols provides said graphical hierarchy.
 342. The graphicalhierarchy representation of claim 341, wherein said first symbolrepresents a node; said additional symbols comprising two symbols thatrespectively represent labels 0 and 1; and wherein said graphicalhierarchy comprises a binary edge labeled tree.
 343. The graphicalhierarchy representation of claim 342, wherein said linear collectionbeing an edge label symbol provided prior to the symbol for a nodeassociated with the edge that it labels.
 344. The graphical hierarchyrepresentation of claim 343, wherein said linear collection being anedge label symbol being provided immediately prior to the symbol for anode associated with the edge that it labels.
 345. The graphicalhierarchy representation of claim 341, wherein said first symbolrepresents an edge; said additional symbols comprising two symbols torespectively represent 0 and 1; and wherein said graphical hierarchycomprises a binary node labeled tree.
 346. The graphical hierarchyrepresentation of claim 345, wherein said linear collection being a nodelabel symbol being provided prior to the symbol for an edge associatedwith for the node that it labels.
 347. The graphical hierarchyrepresentation of claim 346, wherein said linear collection being a nodelabel symbol being provided immediately prior to the symbol for an edgeassociated with the node that it labels.
 348. The graphical hierarchyrepresentation of claim 341, wherein adjacent label symbols representsuccessively connected nodes or edges.
 349. The graphical hierarchyrepresentation of claim 341, wherein non-adjacent label symbolsrepresent nodes or edges that are not successively connected.
 350. Ahierarchical set of data comprising: a string in an enumeration ofstrings having been formed by: positioning at a location k a string suchthat said string corresponds to a binary value, said binary valuecorresponding to k with the left most binary digit omitted, k comprisinga positive natural numeral greater than one.
 351. The hierarchical setof data of claim 350, wherein said strings comprise binary strings. 352.The hierarchical set of data of claim 351, wherein said binary stringscomprise binary edge labeled strings.
 353. The hierarchical set of dataof claim 351, wherein said binary strings comprise binary node labeledstrings.
 354. An ordered set of data comprising: a plurality of treeshaving been ordered by: converting said plurality of trees to numerals;ordering said numerals; and converting said numerals to said pluralityof trees.
 355. The ordered set of data of claim 354, wherein said treescomprise binary edge labeled trees.
 356. The ordered set of data ofclaim 354, wherein said trees comprise binary node labeled trees. 357.The ordered set of data of claim 354, wherein said trees comprisenumeral edge labeled trees, said trees being ordered by: converting saidnumeral edge labeled trees to binary edge labeled trees.
 358. Theordered set of data of claim 354, wherein said trees comprise numeralnode labeled trees, said trees being ordered by: converting said numeralnode labeled trees to binary node labeled trees.
 359. The ordered set ofdata of claim 354, and further comprising a plurality of strings havingbeen ordered by: preceding binary longer strings with shorter binarystrings; and ordering binary strings of the same length based upon theirassociated binary values when a binary one is added to the left of thebinary string; wherein a relationship exists between the ordered treesand the ordered binary strings
 360. The ordered set of data of claim359, wherein said binary strings comprise binary node labeled strings;and wherein ordering binary node labeled strings of the same lengthcomprises ordering binary node labeled strings based at least in partupon the associated binary values.
 361. The ordered set of data of claim359, wherein said binary strings comprise binary edge labeled strings;and wherein ordering binary edge labeled strings of the same lengthcomprises ordering binary edge labeled strings based at least in partupon the associated binary values.
 362. An apparatus comprising: acomputing platform; said computing platform comprising means forconverting at least two trees to numerals, means for multiplying saidnumerals, and means for converting the product of said numerals to atree.
 363. The apparatus of claim 362, wherein said trees comprisebinary edge labeled trees.
 362. The apparatus of claim 362, wherein saidcomputing platform comprises means for converting said at least twotrees to binary edge labeled trees.
 365. The apparatus of claim 362,wherein said at least two trees include at least one null label. 366.The apparatus of claim 365, wherein said computing platform comprisesmeans for converting said at least two trees that include at least onenull label to trees without a null label.
 367. The apparatus of claim366, wherein said computing platform comprises means for convertingtrees without a null type to binary edge labeled trees.
 368. Theapparatus of claim 362, wherein said computing platform comprises meansfor converting said at least two trees to binary node labeled trees.369. The apparatus of claim 362, wherein said numerals comprise binarynumerals.
 370. The apparatus of claim 362, wherein said computingplatform comprises means for converting said numerals to binarynumerals.
 371. An apparatus comprising: a computing platform; saidcomputing platform comprising means for converting at least two trees tostrings, means for combining said strings, and means for converting thecombination of strings to a tree.
 372. The apparatus of claim 371,wherein said trees comprise binary edge labeled trees.
 373. Theapparatus of claim 371, wherein said computing platform comprises meansfor converting said at least two trees to binary edge labeled trees.374. The apparatus of claim 371, wherein said at least two trees includeat least one null label.
 375. The apparatus of claim 374, wherein saidcomputing platform comprises means for converting said at least twotrees that include at least one null label to trees without a nulllabel.
 376. The apparatus of claim 375, wherein said computing platformcomprises means for converting said trees without a null label to binaryedge labeled trees.
 377. The apparatus of claim 371, wherein saidcomputing platform comprises means for converting said at least twotrees to binary node labeled trees.
 378. The apparatus of claim 371,wherein said strings comprise binary strings.
 379. The apparatus ofclaim 378, wherein said binary strings comprise binary edge labeledstrings.
 380. The apparatus of claim 371, wherein said strings comprisebinary strings, and wherein said computing platform comprises means forcombining said binary strings including means for converting said binarystrings to binary numerals, means for multiplying said binary numerals,and means for converting the product to a binary string.
 381. Anapparatus comprising: a computing platform; said computing platformcomprising means for converting a first tree to a first numeral, meansfor factoring said first numeral into at least second and thirdnumerals, and means for converting said at least second and thirdnumerals to at least second and third trees.
 381. The apparatus of claim381, wherein said first tree comprises a binary edge labeled tree. 383.The apparatus of claim 381, wherein said computing platform comprisesmeans for converting said first tree to a binary edge labeled tree. 384.The apparatus of claim 381, wherein said first tree includes at leastone null label.
 385. The apparatus of claim 384, wherein said computingplatform comprises means for converting said first tree to a treewithout a null label.
 386. The apparatus of claim 385, wherein saidcomputing platform comprises means for converting said tree without anull label to a binary edge labeled tree.
 387. The apparatus of claim381, wherein said computing platform comprises means for converting saidfirst tree to a binary node labeled tree.
 388. The apparatus of claim381, wherein said numerals comprise binary numerals.
 389. The apparatusof claim 381, wherein said computing platform comprises means forconverting said numerals to binary numerals.
 390. An apparatuscomprising: a computing platform; said computing platform comprisingmeans for converting a first tree to a first string, means for splittingsaid first string into at least second and third strings, and means forconverting said at least second and third strings to at least second andthird trees.
 391. The apparatus of claim 390, wherein said first treecomprises a binary edge labeled tree.
 392. The apparatus of claim 390,wherein said computing platform comprises means for converting saidfirst tree to a binary edge labeled tree.
 393. The apparatus of claim390, wherein said first tree includes at least one null label.
 394. Theapparatus of claim 393, wherein said computing platform comprises meansfor converting said first tree to a tree without a null label.
 395. Theapparatus of claim 393, wherein said computing platform comprises meansfor converting said tree without a null label to a binary edge labeledtree.
 396. The apparatus of claim 393, wherein said computing platformcomprises means for converting said tree without a null label to abinary node labeled tree.
 397. The apparatus of claim 390, wherein saidcomputing platform comprises means for converting said first tree to abinary node labeled tree.
 398. The apparatus of claim 390, wherein saidstrings comprise binary strings.
 399. The apparatus of claim 398,wherein said computing platform comprises means for converting saidbinary strings to numerals.
 400. The apparatus of claim 398, whereinsaid binary strings comprise binary edge labeled strings.
 401. Theapparatus of claim 390, wherein said strings comprise binary strings,and wherein said means for splitting said first string into at leastsecond and third strings comprises: means for converting said firststring to a binary numeral, means for factoring said binary numeral, andmeans for converting the factors to binary strings.
 402. An apparatuscomprising: a computing platform; said computing platform comprisingmeans for converting a string to a numeral, and means for convertingsaid numeral to a tree.
 403. The apparatus of claim 402, wherein saidstring comprises a binary string.
 404. The apparatus of claim 403,wherein said binary string comprises a binary edge labeled string. 405.The apparatus of claim 402, wherein said tree comprises a binary edgelabeled tree.
 406. The apparatus of claim 402, wherein said treecomprises a binary node labeled tree.
 407. The apparatus of claim 406,wherein said computing platform comprises means for converting saidbinary node labeled tree to a numeral node labeled tree.
 408. Theapparatus of claim 407, wherein said computing platform comprises meansfor converting said numerical node labeled tree to a numeral nodelabeled tree with at least one null label.
 409. The apparatus of claim402, wherein said computing platform comprises means for converting saidstring to a binary edge labeled string.
 410. The apparatus of claim 402,wherein said computing platform comprises means for converting said treeto a tree with at least one null label.
 411. The apparatus of claim 402,wherein said tree comprises a binary edge labeled tree, wherein saidcomputing platform comprises means for converting said binary edgelabeled tree to a numeral edge labeled tree.
 412. The apparatus of claim402, wherein said computing platform comprises means for converting saidbinary edge labeled tree to a numeral edge labeled tree with at leastone null label.
 413. The apparatus of claim 402, wherein said numeralcomprises a binary numeral.
 414. The apparatus of claim 402, whereinsaid computing platform comprises means for converting said numeral to abinary numeral.
 415. An apparatus comprising: a computing platform; saidcomputing platform comprising means for converting a tree to a numeral,and means for converting said numeral to a string.
 416. The apparatus ofclaim 415, wherein said string comprises a binary string.
 417. Theapparatus of claim 416, wherein said binary string comprises a binaryedge labeled string.
 418. The apparatus of claim 415, wherein said treecomprises a binary edge labeled tree.
 419. The apparatus of claim 415,wherein said tree comprises a binary node labeled tree.
 420. Theapparatus of claim 415, wherein said tree comprises a numeral nodelabeled tree, said computing platform comprises means for convertingsaid numeral node labeled tree to a binary node labeled tree.
 421. Theapparatus of claim 415, wherein said tree comprises a numeral nodelabeled tree with at least one null label, said computing platformcomprises means for converting said numeral labeled tree to a numeralnode labeled tree without a null label.
 422. The apparatus of claim 415,wherein said tree comprises a numeral edge labeled tree, wherein saidcomputing platform comprises means for converting said numeral edgelabeled tree to a binary edge labeled tree.
 423. The apparatus of claim415, wherein said numeral comprises a binary numeral.
 424. An apparatuscomprising: a computing platform; said computing platform comprisesmeans for ordering strings, said means for ordering strings comprising:means for preceding longer binary strings with shorter binary strings;and means for ordering binary strings of the same length based at leastupon their associated binary values when a binary one is added to theleft of the binary string.
 425. The apparatus of claim 424, wherein saidbinary strings comprise binary node labeled strings; and wherein saidcomputing platform comprises means for ordering binary node labeledstrings of the same length based at least upon the associated binaryvalues
 426. The apparatus of claim 424, wherein said binary stringscomprise binary edge labeled strings; and wherein said computingplatform comprises means for ordering binary edge labeled strings of thesame length based at least upon the associated binary values.
 427. Anapparatus comprising: a computing platform; said platform comprisingmeans for enumerating trees, said means for enumerating trees comprisingmeans for positioning a tree at location k, where k is the product of uand v, such that said tree is formed by a union of trees in positions uand v; k, u, and v comprising positive natural numerals greater thanone.
 428. The apparatus of claim 427, wherein said trees comprise binaryedge labeled trees.
 429. The apparatus of claim 427, wherein said treescomprise binary node labeled trees.
 430. The apparatus of claim 427,wherein said trees comprise numeral edge labeled trees, wherein saidcomputing platform comprises means for converting said numeral edgelabeled trees to binary edge labeled trees.
 431. The apparatus of claim427, wherein said trees comprise numeral node labeled trees, and whereinsaid computing platform comprises means for converting said numeral nodelabeled trees to binary node labeled trees.
 432. The apparatus of claim427, wherein said computing platform comprises means for positioning atree at location j, where j comprises a non-composite numeral greaterthan one, such that said tree is formed from a prior tree by attaching anode as a root node.
 433. The apparatus of claim 432, wherein saidcomputing platform comprises means for labeling the edge between saidnode and said root node.
 434. The apparatus of claim 433, wherein saidtrees comprise binary edge labeled trees, said means for labeling saidedge comprising means for labeling said edge 0 or
 1. 435. The apparatusof claim 434, wherein said edge capable of being labeled zero, j capableof being P(2m−1), where P(2m−1) is the (2m−1)th non-composite of theKeene enumeration of non-composite numerals on the positive naturalnumerals.
 436. The apparatus of claim 434, wherein said edge capable ofbeing labeled one, j capable of being P(2m), where P(2m) is the (2m)thnon-composite of the Keene enumeration of non-composite numerals on thepositive natural numerals.
 437. The apparatus of claim 427, wherein saidtrees comprise numeral edge labeled trees, and wherein said computingplatform comprises means for converting said numeral edge labeled treesto binary edge labeled trees.
 438. The apparatus of claim 427, whereinsaid computing platform comprising means for positioning at location k astring such that said string corresponds to a binary value, said binaryvalue corresponding to k with the left most binary digit omitted. 439.The apparatus of claim 438, wherein said strings comprise binarystrings.
 440. The apparatus of claim 439, wherein said binary stringcomprises a binary edge labeled string.
 441. The apparatus of claim 439,wherein said binary string comprises a binary node labeled string. 442.An apparatus comprising: a computing platform; said computing platformcomprising means for forming at least one of a zero-push and a one-pushof a tree.
 443. The apparatus of claim 442, wherein said tree comprisesa binary edge labeled tree.
 444. The apparatus of claim 442, whereinsaid computing platform comprises means for forming a tree comprising atleast an N-push of said tree, where N is a positive natural numeralgreater than one.
 445. The apparatus of claim 442, wherein saidcomputing platform comprises means for converting said tree to a binaryedge labeled tree.
 446. The apparatus of claim 445, wherein saidcomputing platform comprises means for converting said tree to saidbinary edge labeled tree from a binary node labeled tree.
 447. Theapparatus of claim 442, wherein said tree comprises a numeral edgelabeled tree
 448. The apparatus of claim 447, wherein said computingplatform comprises means for converting said tree to a binary edgelabeled tree from said numeral edge labeled tree.
 449. The apparatus ofclaim 448, wherein said numeral edge labeled tree includes at least onenull label.
 450. The apparatus of claim 442, wherein said tree comprisesa numeral node labeled tree.
 451. The apparatus of claim 450, whereinsaid computing platform comprises means for converting said tree to abinary edge labeled tree from said numeral node labeled tree.
 452. Theapparatus of claim 450, wherein said numerical node labeled treeincludes at least one null label.
 453. An apparatus comprising: acomputing platform; said computing platform being adapted to employ afirst symbol to represent a node or an edge, and employ additionalsymbols to represent labels of said node or edge, wherein the successiveorder of said symbols provides a graphical hierarchy.
 454. The apparatusof claim 453, wherein said first symbol represents a node; saidadditional symbols comprise two symbols that respectively representlabels 0 and 1; and wherein said graphical hierarchy comprises a binaryedge labeled tree.
 455. The apparatus of claim 454, wherein saidcomputing platform comprises means for providing an edge label symbolprior to the symbol for a node associated with the edge that it labels.456. The apparatus of claim 455, wherein said computing platformcomprises means for providing an edge label symbol immediately prior tothe symbol for a node associated with the edge that it labels
 457. Theapparatus of claim 453, wherein said first symbol represents an edge;said additional symbols comprise two symbols to respectively represent 0and 1; and wherein said graphical hierarchy comprises a binary nodelabeled tree.
 458. The apparatus of claim 457, wherein said computingplatform comprises means for providing a node label symbol prior to thesymbol for an edge associated with the node that it labels.
 459. Theapparatus of claim 458, wherein said computing platform comprises meansfor providing a node label symbol immediately prior to the symbol for anedge associated with the node that it labels.
 460. The apparatus ofclaim 453, wherein said computing platform comprise means for providingadjacent label symbols to represent successively connected nodes oredges.
 461. The apparatus of claim 453, wherein said computing platformcomprises means for providing non-adjacent label symbols to representnodes or edges that are not successively connected.
 462. An apparatuscomprising: a computing platform; said computing platform comprisingmeans for positioning at location k a string such that said stringcorresponds to a binary value, said binary value corresponding to k withthe left most binary digit omitted.
 463. The apparatus of claim 462,wherein said string comprises a binary string.
 464. The apparatus ofclaim 462, wherein said binary string comprises a binary edge labeledstring.
 465. The apparatus of claim 463, wherein said binary stringcomprises a binary node labeled string.
 466. An apparatus comprising: acomputing platform; said computing platform comprising means forconverting a set of trees to numerals, means for ordering said numerals,and means for converting said numerals to said set of trees.
 467. Theapparatus of claim 466, wherein said trees comprise binary edge labeledtrees.
 468. The apparatus of claim 466, wherein said trees comprisebinary node labeled trees.
 469. The apparatus of claim 466, wherein saidtrees comprise numeral edge labeled trees, and wherein said computingplatform comprises means for converting said numeral edge labeled treesto binary edge labeled trees.
 470. The apparatus of claim 466, whereinsaid trees comprise numeral node labeled trees, and wherein saidcomputing platform comprises means for converting said numeral nodelabeled trees to binary node labeled trees.
 471. The apparatus of claim470, wherein said computing platform comprise means for ordering a setof strings, said means for ordering comprising: means for precedinglonger binary strings with shorter binary strings; and means forordering binary strings of the same length based upon their associatedbinary values when a binary one is added to the left of the binarystring; wherein a relationship exists between the ordered trees and theordered binary strings.
 472. The apparatus of claim 471, wherein saidbinary strings comprise binary node labeled strings; and wherein saidcomputing platform comprises means for ordering binary node labeledstrings of the same length based at least in part upon the associatedbinary values.
 473. The apparatus of claim 471, wherein said binarystrings comprise binary edge labeled strings; and wherein said computingplatform comprises means for ordering binary edge labeled strings of thesame length based at least in part upon the associated binary values.